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Proofs of Trigonometric Identities

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Verifying a Trigonometric Identity

Verify that \frac{\sin^2x}{\tan^2x}=1 - \sin^2x .

Guidance

This concept continues where the previous one left off. Now that you are comfortable simplifying expressions, we will extend the idea to verifying entire identities. Here are a few helpful hints to verify an identity:

  • Change everything into terms of sine and cosine.
  • Use the identities when you can.
  • Start with simplifying the left-hand side of the equation, then, once you get stuck, simplify the right-hand side. As long as the two sides end up with the same final expression, the identity is true.

Example A

Verify the identity \frac{\cot^2x}{\csc x}=\csc x - \sin x .

Solution: Rather than have an equal sign between the two sides of the equation, we will draw a vertical line so that it is easier to see what we do to each side of the equation. Start with changing everything into sine and cosine.

\begin{array}{c|c c} \frac{\cot^2x}{\csc x} & \csc x - \sin x \\\frac{\frac{\cos^2x}{\sin^2x}}{\frac{1}{\sin x}} & \frac{1}{\sin x}- \sin x \\\frac{\cos^2x}{\sin x}\end{array}

Now, it looks like we are at an impasse with the left-hand side. Let’s combine the right-hand side by giving them same denominator.

\begin{array}{|c} \frac{1}{\sin x}- \frac{\sin^2x}{\sin x} \\\frac{1- \sin^2x}{\sin x} \\\frac{\cos^2x}{\sin x}\end{array}

The two sides reduce to the same expression, so we can conclude this is a valid identity. In the last step, we used the Pythagorean Identity, \sin^2 \theta+\cos^2 \theta=1 , and isolated the \cos^2x=1- \sin^2x .

There are usually more than one way to verify a trig identity. When proving this identity in the first step, rather than changing the cotangent to \frac{\cos^2x}{\sin^2x} , we could have also substituted the identity \cot^2x=\csc^2x-1 .

Example B

Verify the identity \frac{\sin x}{1- \cos x}=\frac{1+ \cos x}{\sin x} .

Solution: Multiply the left-hand side of the equation by \frac{1+ \cos x}{1+ \cos x} .

\frac{\sin x}{1- \cos x}&= \frac{1+ \cos x}{\sin x} \\\frac{1+ \cos x}{1+ \cos x} \cdot \frac{\sin x}{1- \cos x}&= \\\frac{\sin \left(1+\cos x\right)}{1- \cos^2x}&= \\\frac{\sin \left(1+\cos x\right)}{\sin^2x}&= \\\frac{1+\cos x}{\sin x}&=

The two sides are the same, so we are done.

Example C

Verify the identity \sec(-x)=\sec x .

Solution: Change secant to cosine.

\sec(-x)= \frac{1}{\cos \left(-x\right)}

From the Negative Angle Identities, we know that \cos (-x)=\cos x .

&=\frac{1}{\cos x} \\&=\sec x

Concept Problem Revisit Start by simplifying the left-hand side of the equation.

\frac{\sin^2x}{\tan^2x}=\frac{\sin^2x}{\frac{\sin^2x}{\cos^2x}}\\=\cos^2x .

Now simplify the right-hand side of the equation. By manipulating the Trigonometric Identity,

\sin^2x + \cos^2x = 1 , we get \cos^2x = 1 - \sin^2x .

\cos^2x =\cos^2x and the equation is verified.

Guided Practice

Verify the following identities.

1. \cos x \sec x=1

2. 2- \sec^2x=1- \tan^2x

3. \frac{\cos \left(-x\right)}{1+ \sin \left(-x\right)}=\sec x+ \tan x

Answers

1. Change secant to cosine.

\cos x \sec x&=\cos \cdot \frac{1}{\cos x} \\&=1

2. Use the identity 1+ \tan^2 \theta=\sec^2 \theta .

2- \sec^2x&=2-(1+ \tan^2x) \\&=2-1- \tan^2x \\&=1- \tan^2x

3. Here, start with the Negative Angle Identities and multiply the top and bottom by \frac{1+ \sin x}{1+ \sin x} to make the denominator a monomial.

\frac{\cos \left(-x\right)}{1+ \sin \left(-x\right)}&=\frac{\cos x}{1- \sin x} \cdot \frac{1+ \sin x}{1+ \sin x} \\&=\frac{\cos x \left(1+ \sin x \right)}{1- \sin^2x} \\&=\frac{\cos x \left(1+ \sin x\right)}{\cos^2x} \\&=\frac{1+ \sin x}{\cos x} \\&=\frac{1}{\cos x}+ \frac{\sin x}{\cos x} \\&=\sec x+ \tan x

Practice

Verify the following identities.

  1. \cot (-x)=- \cot x
  2. \csc (-x)=- \csc x
  3. \tan x \csc x \cos x=1
  4. \sin x+ \cos x \cot x=\csc x
  5. \csc \left(\frac{\pi}{2}-x\right)=\sec x
  6. \tan \left(\frac{\pi}{2}-x\right)=\tan x
  7. \frac{\csc x}{\sin x}- \frac{\cot x}{\tan x}=1
  8. \frac{\tan^2x}{\tan^2x+1}=\sin^2x
  9. (\sin x- \cos x)^2+(\sin x+ \cos x)^2=2
  10. \sin x- \sin x \cos^2x= \sin^3x
  11. \tan^2x+1+\tan x \sec x=\frac{1+ \sin x}{\cos^2x}
  12. \cos^2x=\frac{\csc x \cos x}{\tan x+ \cot x}
  13. \frac{1}{1- \sin x} - \frac{1}{1+ \sin x}=2 \tan x \sec x
  14. \csc^4x- \cot^4x=\csc^2x+\cot^2x
  15. (\sin x - \tan x)(\cos x- \cot x)=(\sin x-1)(\cos x-1)

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